Are $(x ^ 2, xy)$ and $(x) ∩ (x ^ 2, xy, y ^ 2)$ of $k [x, y]$ the same ideals in $k[x,y]$?

Question

Let $k$ be a field. Are $(x ^ 2, xy)$ and $(x) ∩ (x ^ 2, xy, y ^ 2)$ of $k [x, y]$ the same ideals? The hint says $(x ^ 2, xy, y ^ 2)$ is the same as the subset of $k [x, y]$ of degree with 2 or higher, and $(x ^ 2, xy)$ is not a prime ideal. But I do not know how to apply these hints to this problem. Thank you.

Answer 1

Let $I=(x^2,xy)$. Clearly, $$I\subseteq (x)\cap (x^2,xy,y^2).$$ Now take $f\in (x)\cap (x^2,xy,y^2)$. Since $f\in(x)$, every monomial of $f$ is a multiple of $x$. I claim that there is no monomial term of the form $cx$ where $c\in k$. The sum of degree $\geq 2$ monomials of $f$ forms a polynomial $g$ which belongs to the ideal $I=(x^2, xy)$. So we can write $f=g+cx$. If $c\neq 0$, then $$x=c^{-1}cx=c^{-1}(f-g)\in [(x)\cap (x^2,xy,y^2)]+I\subset (x^2,xy,y^2),$$ but this is impossible.